There appears to be an error in the influential paper by Rahmstorf et al. (2007). Rahmstorf et.al. (Science Brevia, 4 May 2007, p709 [1]) reports that the trend of the global mean of surface temperature and sea level raise concerns that the climate system “may be responding more quickly to climate change than our current generation of models indicates”. At least one major study, Interim Report of the Garnaut Review, relies on the paper for advocating prompt and extreme action on carbon emissions, one of its major conclusions (Section 2.4 Consequences of Climate Change, Observed Climate Change). But there seems to be a problem.

As previously reported here,
the conclusions of Rahmstorf’s 7 (Rahmstorf, Cazenave, Church, Hansen, Keeling, Parker, and Somerville) rely on a trend line lying above the IPCC projections on their Figure 1, shown enlarged below. No statistical tests are performed, the basis for their claim is purely based on the visual aid. Their Figure 1 is below, with the key part of the image containing the IPCC projection enlarged.



Figures 1 and 2: Rahmstorf et al. (2007) Figure 1. Whole and enlarged.

Rahmstorf’s 7 state in the figure caption that
“All trends are nonlinear trend lines
and are computed with an embedding period of 11 years
and a minimum roughness criterion at the end (Moore
et.al. 2006 [2])”. On reading Moore’s paper, it would
appear the nonlinear methodology used was Singular
Spectrum Analysis (SSA). The Moore paper suggests the
minimum roughness criterion (MRC) would follow the
Mann 2004 [3] recipe of padding the end of the series with
data reflected about the final value.



Figure 3 and 4: GISS temperature with SSA trend, unpadded and padded.

The peculiar property of MRC of “pinning” the trend line
to the final end point of the series has been noted in
a post “Mannomatic Smoothing and Pinned End-points” at ClimateAudit.

The comparison of MRC and non-MRC padded series
is shown in figures captured from CaterpillarSSA.
The first figure “unpadded.png” shows
a SSA trend line that approximates the Rahmstorf figure result. The
second figure “padded.png” shows the SSA trend line with an MRC
padded GISS series, passing directly through the 2006
value. This is as it should be, as the MRC effectively
‘pins’ the trend line to the final value due to the
symmetry about the final value.

Was a direct application
of the SSA trend line used and not an MRC padded series
as described?

If an MRC padded series had been used in the figure,
it would have been end-pinned to the 2006 value,
at the center of the IPCC
projections. The figure would then not have conveyed the
impression that temperatures are in the upper range of
the IPCC projections, as claimed.

As it was in 2006, it appears that SSA
without MRC padding produces a higher trend line than
with MRC padding, necessary for supporting their claim.

An additional puzzling factor is the reference to MRC padding
at all. Padding the end of the series is not actually
necessary to ensure a SSA trend line is drawn to the
end of the series values. Padding is only necessary in
acausal filters such as moving averages that stop a
window length short of the end of the series.

To date, attempts to contact Prof. Rahmstorf and clarify the actual methodology
used have been unsuccessful.


[1]Stefan Rahmstorf, Anny Cazenave, John A. Church, James E. Hansen,
Ralph F. Keeling, David E. Parker, and Richard C. J. Somerville. Recent
Climate Observations Compared to Projections. Science, 316(5825):709–,

[2] A. Grinsted Moore, J.C. and S. Jevrejeva. New tools for analyzing time series relationships and trends. Eos, 86(24), 1995.

[3] M. E. Mann. On smoothing potentially non-stationary climate time series.
Geophys. Res. Lett., 31:L07214, doi:10.1029/2004GL019569 2004.

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